But why should we need such an indirect route to a Lagrangian formulation? Recall that in classical mechanics the Hamiltonian and Lagrangian viewpoints are often used interchangeably, largely because it is often easy to perform the Legendre transform which transitions between the Hamiltonian picture based on the cotangent bundle and the Lagrangian picture based on the tangent bundle (see e.g., [1]}). In [2]}, however, A. Caticha does, in fact, model the e-phase space as a cotangent bundle. Should this not suggest a corresponding Lagrangian formulation of ED? In short, yes. Nevertheless, while this route may be available in principle, in practice obtaining such a Lagrangian picture would require us to invert the local differential FP equation to obtain the equivalent of \(p = \partial L /\partial \dot{q}\)  in classical mechanics, which would here be a non-local integral equation. Thus suggests that the Hamiltonian viewpoint, being more simple and compact, should actually be considered the more fundamental one.
